Question

В прямоугольный треугольник с катетами 6 см и 10 см вписан имеющий с ним общий угол прямоугольник наибольшей площади. Найти площадь прямоугольника. (Решать через функцию)

Answer 1

1. Достраиваем исходный прямоугольный треугольник до прямоугольника.
2. Проводим вторую диагональ получившегося прямоугольника.
3. Получилось четыре одинаковых прямоугольных треугольника.
4. Разбиваем прямоугольник на четыре равных прямоугольника проводя параллельные прямые через точку пересечения диагоналей.
5. Получившиеся прямоугольники имеют наибольшую площадь так как в сумме дают полную площадь прямоугольника.
6. Площадь прямоугольника 8*5=40 см².
7. Площадь вписанного прямоугольника 40/4=10 см².

Answer 2

Finding the Area of the Rectangle Inscribed in a Right Triangle

To find the area of the rectangle inscribed in a right triangle, we can use the following steps:

1. Determine the lengths of the two legs (or catheti) of the right triangle. In this case, the lengths are given as 6 cm and 10 cm.

2. Identify the right angle of the right triangle. This is the angle formed by the intersection of the two legs.

3. Draw a rectangle inside the right triangle, such that one side of the rectangle is aligned with one of the legs of the right triangle, and the other side of the rectangle is aligned with the other leg of the right triangle.

4. The rectangle with the largest area that can be inscribed in the right triangle will have its vertices at the midpoints of the hypotenuse and the two legs of the right triangle.

5. To find the area of the rectangle, we can use the formula: Area = length × width.

6. The length of the rectangle is equal to the length of the hypotenuse of the right triangle, which can be found using the Pythagorean theorem: c = sqrt(a^2 + b^2), where a and b are the lengths of the legs of the right triangle.

7. The width of the rectangle is equal to half the length of one of the legs of the right triangle.

Let's calculate the area of the rectangle using the given lengths of the legs (6 cm and 10 cm):

1. Length of the hypotenuse (c) = sqrt(6^2 + 10^2) = sqrt(36 + 100) = sqrt(136) cm.

2. Width of the rectangle = 6 cm / 2 = 3 cm.

3. Area of the rectangle = length × width = sqrt(136) cm × 3 cm = 3 sqrt(136) cm^2.

Therefore, the area of the rectangle inscribed in the right triangle with legs of 6 cm and 10 cm is 3 sqrt(136) cm^2.

Please note that the above calculations are based on the assumption that the rectangle with the largest area is inscribed in the right triangle. If you have any further questions or need clarification, feel free to ask!